Driven oscillating nonlinear acoustic waves

A computational investigation has been conducted for a weakly nonlinear model of acoustical plane wave propagation in a cylindrical tube. At the left end a harmonic driving actuator excites standing waves in the tube with a solid wall at the right hand end. For driving frequencies a little higher than the first resonance frequency of the simple wave equation, neglecting nonlinearities and dissipation, we observe formation of oscillating shock waves in the weakly nonlinear model. A little above the second harmonic frequency of the simple linear wave equation we observe two counter propagating shock waves in the tube. Increasing the driving frequency further the oscillation mode turns into two parallel propagating shocks. A Hamiltonian energy analysis is included, which demonstrates balance between the energy input from the driver and the dissipation of the wave dynamics.